Metric Spaces Introduction, Real Analysis II
In this lecture, I define the concept of a metric space, a fundamental domain in real analysis. A metric space requires two things: a set M and a function D that measures the distance between elements in M. We'll explore the four essential properties a metric must satisfy: positivity, non-degeneracy, symmetry, and the triangle inequality.
(MA 426 Real Analysis II, Lecture 4)
We begin by defining these properties in detail. Positivity means the distance is never negative. Non-degeneracy ensures the distance is zero if and only if the elements are the same. Symmetry indicates that the distance from X to Y is the same as from Y to X. The triangle inequality states that the direct distance between two points is always less than or equal to the distance traveled via a third point.
To illustrate, we explore a discrete metric space, where the distance as either 0 or 1. We'll verify this space meets all metric properties. Next, we examine the taxi cab metric on the xy-plane, calculating distances based on city block paths rather than straight lines. We'll check that this metric also satisfies the four properties and discuss its unique geometric implications, such as diamond-shaped circles.
Finally, I demonstrate that any normed vector space induces a metric, proving that the norm's properties ensure the metric properties. This concludes with a discussion on the broader implications and limitations of metrics derived from norms.
#advancedcalculus #MathLecture #MetricSpace #RealAnalysis #Mathematics #MathProof #DiscreteMetric #TaxiCabMetric #EuclideanSpace #Geometry #NormedVectorSpace #MathConcepts
Watch video Metric Spaces Introduction, Real Analysis II online without registration, duration hours minute second in high quality. This video was added by user Dr. Bevin Maultsby 12 August 2024, don't forget to share it with your friends and acquaintances, it has been viewed on our site 347 once and liked it 20 people.